An electron in a one-dimensional box has ground-state energy 0.010 eV. (a) What is the length of

An electron in a one-dimensional box has ground-state energy...

Key Concepts

Particle in a Box Model

This model represents a quantum system where a particle is confined in a region with infinitely high potential barriers at the boundaries. In this simplified one-dimensional system, the particle can only occupy discrete energy levels. The model is fundamental in quantum mechanics as it illustrates how spatial confinement leads to the quantization of energy and the formation of standing wave patterns for the particle's wave function.

Quantization of Energy Levels

In a confined quantum system like the particle in a box, the energy levels are not continuous but are instead quantized. The allowed energies depend on quantum numbers and are determined by the boundary conditions of the system. This quantization is typically represented by the expression E_n = (n^2 ?^2 ?^2) / (2mL^2), where n is a positive integer, reflecting the discrete nature of the energy states due to the confinement.

Wave Functions and Standing Waves

The wave functions in a particle in a box problem describe the probability amplitude of finding the particle within the box. They take the form of sine functions, which naturally satisfy the zero probability conditions at the boundaries of the box. The number of nodes in the wave function increases with the energy level, illustrating the concept of standing waves in confined systems.

de Broglie Wavelength

The de Broglie wavelength associates a wave-like nature to particles. In a confined system, the de Broglie wavelength is related to the momentum of the particle and thus to its energy. The wavelength concept helps in understanding the spatial periodicity and interference patterns of the wave function within the potential well, and it is crucial for interpreting phenomena such as diffraction and transition probabilities.

Quantum Transitions and Photon Emission/Absorption

When a quantum system transitions between energy states, it either absorbs or emits a photon whose energy corresponds to the difference between those energy levels. This concept is central to understanding how electrons in confined systems interact with electromagnetic radiation. The wavelength of the emitted or absorbed photon can be determined by equating its energy (using the Planck relation, E = hc/?) to the difference in quantized energy states of the system.

Transcript

00:01 In this exercise, we have an electron that is confined to a one -dimensional box, and we have the information that the ground state energy of the electron is equal to 0 .01 electron volt.

00:14 In question a, we have to find what is the length of the box.

00:22 So what i'm going to do is to write the ground state energy as h squared divided by 8ml squared, this is the formula for the ground state energy.

00:36 And if we want to find l, i'm going to isolate l here.

00:40 So l squared is equal to h squared divided by 8me1.

00:47 And for that reason, l is equal to h divided by the square root of 8me1.

00:59 So l is equal to 6 .63 times 10 to the minus 2 .3.

01:05 24 joules second divided by the square root of 8 times the mass of the electron, which is 9 .1 times 10 to the minus 31, times the ground state energy, which is 0 .01 electron volts.

01:23 And i'm going to multiply this by 1 .6 times 10 to the minus 19 joules per electron volt, just to convert the energy unit.

01:35 So l is equal to 6 .14 times 10 to the minus 9 meters, which is the same as 6 .14 nanometers.

01:53 In question b, we have to sketch the wave functions of the three states that have the lowest energies.

02:01 These states are the ground state and equals 1.

02:05 The first excited state and equals 2.

02:09 And the second excited state n equals 3.

02:14 And we know i'm going to do first for n equals 1.

02:18 So here i have l.

02:20 Here i have 0, which is the first wall of the box.

02:28 We know that for all of these cases, the wave function vanishes at the walls.

02:37 And for the ground state, there are no nodes of the wave function.

02:42 Function in between the walls and the wave function looks like a sign.

02:50 And for each subsequent excited state, there is one additional node in the wave function.

02:58 So this here is for n equals 1, the ground state.

03:05 Now i'm going to sketch for n equals 2, that is the first excited state, where there should be one node.

03:19 So it's one node and then the wave function vanishes.

03:24 This is the wave function for n equals 2.

03:28 And for n equals 3, there are two nodes.

03:37 So basically the way function starts at 0, then there are 1, 2, and 3 nodes.

03:49 And then, i'm sorry, there are 2 nodes, of course, not 3.

03:54 Because n equals 2 is the, n equals 3 is the second excited state.

03:59 So there's one and two nodes, and this is a sketch of the wave function for n equals 3.

04:15 Now for question c, we have to find what is the wavelength of the electron for the second excited state.

04:30 Notice that the wavelength can be read from the graph that we have just drawn.

04:39 So notice that one wavelength is the distance between two consecutive nodes of the wave function.

04:50 So this here is the wavelength.

04:58 And in total for the second excited state, we have the distance.

05:05 Sorry, this is the wavelength, not l.

05:07 This is the wavelength.

05:10 And for the second excited state here, we have n equals 3.

05:21 We have that the distance l is equal to three halves of the wavelength...

Please give Ace some feedback